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In atomic physics, the Hund rule says that the largest spin and orbital stateis realized due to the interplay of the spin-orbit coupling (SOC) and theCoulomb interactions. Here, we show that in ferromagnetic solids the effectiveSOC and the orbital magnetic moment can be dramatically enhanced by a factor of$1/[1-(2U^\prime-U-J_H)\rho_0]$, where $U$ and $U^\prime$ are the on-siteCoulomb interaction within the same oribtals and between different orbitals,respectively, $J_H$ is the Hund coupling, and $\rho_0$ is the average densityof states. This factor is obtained by using the two-orbital as well asfive-orbital Hubbard models with SOC. We also find that the spin polarizationis more favorable than the orbital polarization, being consistent withexperimental observations. This present work provides a fundamental basis forunderstanding the enhancements of SOC and orbital moment by Coulombinteractions in ferromagnets, which would have wide applications inspintronics.

Enhanced spin-orbit coupling and orbital moment in ferromagnets by electron correlations